NUMERICAL METHODS FOR ELECTROMAGNETIC FIELDS AND CIRCUITS

Course Description

The course provides the theoretical and methodological tools for the numerical analysis of electromagnetic fields and for the modelling of electromagnetic structures using spatial and space-time discretization techniques. After reviewing the fundamentals of classical electromagnetism, the main numerical methods for solving field problems are introduced, with particular emphasis on the Finite Element Method (FEM), the Finite Difference Method (FDM), the Boundary Element Method (BEM), and hybrid methods. The course also addresses applications in computational electromagnetics, the use of electromagnetic CAD software, and transmission line theory, providing students with the tools required to address design and simulation problems in electrical and electronic engineering.

Learning Outcomes

At the end of the course, students will be able to:

Knowledge and Understanding

• Describe the theoretical foundations of classical electromagnetism and the main formulations of field problems.
• Understand the mathematical principles underlying numerical methods for spatial and space-time discretization.
• Understand the theoretical foundations of the main numerical methods for solving field problems (FDM, FEM, BEM, and hybrid methods).
• Understand the variational and differential formulations employed in the discretization of electromagnetic problems.
• Understand transmission line models and the main applications of computational electromagnetics.

Applying Knowledge and Understanding

• Formulate mathematical models of electromagnetic problems using differential and integral equations.
• Apply the main numerical methods (FDM, FEM, BEM, and hybrid methods) to the solution of field problems.
• Use CAD tools for the simulation and analysis of electromagnetic devices and structures.
• Analyse transmission lines in the time domain, frequency domain, and under sinusoidal steady-state conditions.
• Critically interpret numerical results by assessing their accuracy, convergence, and limitations.
• Critically interpret numerical results by assessing their accuracy, limitations, and physical significance.

Making Judgements

• Select the most appropriate numerical method according to the problem under analysis.
• Critically assess the reliability of the numerical solutions obtained.
• Analyse the causes of possible discrepancies between theoretical results and numerical simulations.

Communication Skills

• Use the terminology of computational electromagnetics correctly.
• Present mathematical models, numerical procedures, and simulation results in written and oral form.
• Communicate methodologies, results, and limitations of electromagnetic analysis techniques using appropriate technical language.

Learning Skills

• Develop analytical and problem-solving skills in the modelling of electromagnetic problems.
• Develop autonomy in the use of numerical tools, CAD environments, and electromagnetic simulation software.
• Consolidate a study method suitable for addressing advanced topics in computational electromagnetics and computer-aided design. 

The course consists of lectures devoted to the presentation of the theoretical foundations of computational electromagnetics and the main numerical methods for field analysis (Theoretical Teaching), complemented by numerical exercises and laboratory activities using electromagnetic CAD software (Interactive Teaching). The exercises enable students to apply the methods studied to the solution of field problems and to the analysis of transmission lines, thereby developing the skills and competencies specified in the learning outcomes.

If the teaching is taught in mixed or remote mode, the necessary variations may be introduced with respect to what was previously declared, in order to respect the planned program and reported in the syllabus.

The lessons of prof. Laudani are related to "Computational electromagnetism."

- Electromagnetic CAD (Laboratory 6 hours)

3. Electromagnetism.

- Electric charge and its properties.
- Stationary electric field. Global properties and local properties of the electric field in vacuum. Electric scalar potential. Electric field and electric potential of a charge distribution. Electric field in conductors. Electrostatic induction. Capacity coefficients and potential coefficients, proper and partial, of a system of conductors; electrostatic screens. Equivalent network of capacitors. Capacitor. Capacity calculation examples. Electric field energy. Mechanical stresses on charges and conductors. Polarization of a dielectric, polarization intensity vector. Electric field in matter; electric displacement vector. Form and name of constitutive equations. Notes on dissipative phenomena in dielectrics: loss angle, corona effect; dielectric strength and discharge phenomena in dielectrics.
- Stationary current field. Conduction, convection, advection, diffusion electric current; electromotive force generators. Current density vector. Global properties and local properties of the current field. Form and name of constitutive equations; Ohm's law for specific quantities, electromotive field. Calculation of the steady-state current field in linear media. Joule's law for specific quantities and energy balance. Conductances and resistances, proper and partial, of a system of conductors; equivalent network of resistors. Examples of resistance calculations.
- Stationary magnetic field. Global properties and local properties of the magnetic field in vacuum. Poisson vector equation; Biot-Savart law. Magnetic scalar potential. Magnetic field energy. Inductance matrix of a conductor system. Internal inductance and external inductance of a conductor. Examples of inductance calculations. Magnetic field in matter; magnetization intensity vector. Form and name of constitutive equations. Diamagnetic, paramagnetic and ferromagnetic media. First magnetization curve; normal, incremental and differential permeability; hysteresis loop, permanent magnets. Magnetic circuits. Flow tube. Hopkinson's law, reluctance, loss figure, equivalent electrical circuit. Mechanical stresses on the conductors.
- Quasi-stationary magnetic field. Equation of the diffusion of the magnetic field in conductors; skin and proximity effect, depth of penetration; conducting half-space, conducting plate, conductor of circular section. Induced currents.
- Electromagnetic field. Global properties, for fixed and mobile domains, of the electromagnetic field. Faraday-Neumann-Lenz law; dynamic and motivational induced electromotive force. Ampere-Maxwell law; displacement current. Local properties of the electromagnetic field; Maxwell's equations, interface conditions and regularity conditions at infinity. Electromagnetic potentials; auxiliary (gauge) conditions. Wave equations of potentials. Energy and mechanical stresses of the electromagnetic field; vector and Poynting theorem. Uniform electromagnetic waves; propagation of plane waves and spherical waves in material media. Sinusoidal electromagnetic field. Helmholtz equation; solution, boundary conditions and radiation condition. Monochromatic plane waves; dispersion relation, types of polarization. Vector and complex Poynting theorem.
- Deduction of the circuit model of an electrical system. Kirchhoff's laws.

- Elements of electromechanical energy conversion.

4. Transmission lines.

- Transmission Line Model. Free propagation and guided propagation; distributed parameter systems and propagation modes. Assumptions, deductions and validity limits of the Transmission Line model. Two-conductor lines, uniform lines, non-distorting lines, ideal lines. Calculation of the primary parameters of a coaxial line.
- Analysis in the time domain. System of first order equations, second order equations; initial conditions and boundary conditions. Energy balance of a line. Study of non-distorting lines and ideal lines; group velocity and phase velocity; form, properties and physical interpretation of the solutions. Elementary case studies of ideal lines with bipolar terminations. Notes on multi-conductor lines.
- Analysis in the pulsation domain. Line equations in the pulsation domain; propagation parameter and characteristic impedance. Form, properties and physical interpretation of solutions. Impedance, admittance, reflection coefficient, transmission coefficient. Representations of immittance, transmission, diffusion. Study of non-distorting lines of finite length. Adapted line. Elementary case studies of ideal lines with resistive terminations.
- Analysis of lines in sinusoidal regime. Deduction of the equations and properties of the lines in the sinusoidal regime from the analogues valid in the pulsation domain. Study of non-distorting lines; voltage trend and current trend, VSWR, lines at λ/2, lines at λ/4. Elementary case studies of ideal lines with bipolar terminations. Energy balance. Smith diagram and use of it; fitting a line using stubs.

Contribution of the course to the Goals of the 2030 Agenda for Sustainable Development
The topics covered in the course and the acquired knowledge are directly or indirectly aimed at the development of sustainable technological solutions, as well as contributing to a high quality education, in accordance with Goals 4, 7, 9, 11, 12, 13 of the 2030 Agenda for Sustainable Development.

1.	P. P. Silvester, R. L. Ferrari: “Finite elements for electrical engineers”, 3rd edition, Cambridge University Press, 2003.
2.	Jian-Ming Jin: The Finite Element Method in Electromagnetics, 3rd Edition, Wiley-IEEE Press, 2014
3.	S. Alfonzetti: " Lecture Notes of the Course on Numerical Methods".
4.	A. Laudani: " Lecture Notes of the Course on Numerical Methods".

Note: Information for students with disabilities and/or specific learning disorders (SLD)

To ensure equal opportunities and in compliance with current regulations, interested students may request a personal meeting to plan any necessary compensatory measures and/or dispensations, based on educational objectives and specific needs. Students may also contact the designated CInAP representative.